UDC 519.48

MATHEMATICS

I. L. KANTOR

ON INFINITE-DIMENSIONAL SIMPLE GRADED LIE ALGEBRAS

(Presented by Academician I. G. Petrovsky on 22 V 1967)

A Lie algebra \(U\) is called graded if \(U=\sum_{-\infty}^{\infty} U_i\) (direct sum), where the \(U_i\) are finite-dimensional subspaces such that \([U_i,U_j]\subset U_{i+j}\). In the present note the ground field is assumed to be the field of complex numbers, but the greater part of the results is valid for an arbitrary field of characteristic zero.

We shall consider simple graded algebras satisfying the additional condition

\[ [U_{-1},U_{-k}]=U_{-(1+k)}\quad (k>0). \tag{1} \]

First of all we give the construction of an infinite-dimensional simple graded algebra \(\widetilde U\), which is in a certain sense “universal” (see Theorem 2). In the algebra \(\widetilde U\) the subalgebra \(\widetilde U_-=\sum_{i=1}^{-\infty}\widetilde U_i\) is the free Lie algebra generated by the \(n\)-dimensional vector space \(E_n=\widetilde U_{-1}\), with its natural grading. In other words, the elements of \(\widetilde U_{-k}\) are linear combinations of elements \(a_1\circ a_2\circ\cdots\circ a_k\), where \(\circ\) is the operation in the free Lie algebra and \(a_i\in E_n\). Further, by definition, for \(k\ge 0\) the space \(\widetilde U_k\) is \(\operatorname{Hom}(E_n,\widetilde U_{k-1})\), and \(\widetilde U_0=\operatorname{Hom}(E_n,E_n)\), i.e. the space \(\widetilde U_k\) is the space of all \((k+1)\)-linear operators \(A(x_1,x_2,\ldots,x_{k+1})\) acting in \(E_n\) with values in \(E_n\).

We complete the definition of the commutation operation on the space \(\widetilde U\) obtained as follows. Let \(A_k=A_k(x_1,x_2,\ldots,x_k)\in \widetilde U_{k-1}\) and \(A_l=A_l(x_1,x_2,\ldots,x_l)\in \widetilde U_{l-1}\). Then

\[ [A_k,A_l]=A_k\square A_l-A_l\square A_k, \tag{2} \]

where

\[ A_k\square A_l(x_1,x_2,\ldots,x_{k+l-1})= \]

\[ =\sum_{s=0}^{k-1}\ \sum_{i_1<i_2<\cdots<i_s}^{\,l+s-1} A_k\bigl(x_{i_1},x_{i_2},\ldots,x_{i_s}, A_l(x_1,x_2,\ldots,\widehat{x}_{i_1},\ldots,\widehat{x}_{i_s},\ldots,x_{l+s}), \]

\[ x_{l+s+1},\ldots,x_{k+l-1}\bigr); \tag{3} \]

here \(\sum_{i_1<i_2<\cdots<i_s}^{\,l+s-1}\) denotes summation over all ordered sets of \(s\) indices, each of which is not greater than \(l+s-1\), and the symbol \(\widehat{x}_{i_p}\) means that in the sequence \(x_1,x_2,\ldots\) the term with number \(i_p\) is omitted.

To define the commutation operation between \(\widetilde U_{-k}\) and \(\widetilde U_l\), note that every element of the free Lie algebra \(a_1\circ a_2\circ\cdots\circ a_k\) can be written as a sum of elements of the form \(a_{i_1}a_{i_2}\cdots a_{i_k}\), belonging to the free associative algebra generated by the space \(E_n\) (for example, \(a_1\circ a_2=a_1a_2+a_2a_1\)). We define the commutation operation between an \((l+1)\)-linear operator \(A(x_1,x_2,\ldots,x_{l+1})\) and an associative element of the \(k\)-th

of order \(b \equiv b_1 b_2 \ldots b_k\) by the formula

\[ [A,b]= \begin{cases} A(b_1,b_2,\ldots,b_k,x_1,x_2,\ldots,x_{l-k+1}), & \text{if } l+1\ge k,\\ A(b_1,b_2,\ldots,b_{l+1})\circ b_{l+2}\circ\cdots\circ b_k, & \text{if } l+1<k. \end{cases} \tag{4} \]

Thus the commutation operation in the algebra \(\widetilde U\) is completely determined.

Theorem 1. The algebra \(\widetilde U\) is a simple Lie algebra.

Theorem 2. Let \(U\) be a graded algebra satisfying condition (1), containing no ideals belonging to \(U_+=\sum_{i=0}^{\infty}U_i\). Let \(\widetilde U\) be the Lie algebra defined above, for which \(\dim \widetilde U_{-1}=\dim U_{-1}\). Then there exist: 1) a graded subalgebra \(U^*\) of the algebra \(\widetilde U\), containing \(\widetilde U_-\); 2) a graded ideal \(D\subset \sum_{i=-2}^{-\infty}\widetilde U_i\) of the subalgebra \(U^*\) such that \(U^*/D=U\).

The proof of Theorem 2 follows from the following considerations.

Lemma 1. Let \(M\) be an arbitrary Lie algebra. Denote

\[ P_a^k(x_1,x_2,\ldots,x_k)\equiv[\ldots[[a,x_1]x_2],\ldots,x_k], \]

where \(a,x_i\in M\). Then

\[ P_{[a,b]}^{k+l-1}=P_a^k \square P_b^l-P_b^l \square P_a^k, \tag{5} \]

where the operation \(\square\) is defined by formula (3).

Lemma 2. Let \(U\) be a graded algebra satisfying condition (1), containing no ideals belonging to \(U_+=\sum_{i=0}^{\infty}U_i\). Assign to each \(a\in U_k\) \((k\ge0)\) the \(k+1\)-linear operator \(F(a)\) on \(U_{-1}\):

\[ a \xrightarrow{F} [\ldots[[a,x_1],x_2],\ldots,x_{k+1}]. \]

Under the mapping \(F\), the subalgebra \(U_+=\sum_{i=0}^{\infty}U_i\) is isomorphically embedded in the subalgebra \(\widetilde U_+=\sum_{i=0}^{\infty}\widetilde U_i\).

We now denote by \(U_+^F\) the image of the algebra \(U_+\) under the mapping \(F\), and by \(D\) the graded ideal of the subalgebra \(\widetilde U_-\) such that \(U_-=\widetilde U_-/D\). It turns out that the space \(\widetilde U_-+U_+^F\) is a subalgebra \(U^*\) of the algebra \(\widetilde U\), and the ideal \(D\) is carried into itself by elements from \(U_+^F\). This proves Theorem 2.

Let us give examples illustrating Theorem 2. In these examples the subalgebra \(U^*\) coincides with \(N(D)\), the normalizer of \(D\) in the algebra \(\widetilde U\).

The Lie algebra of all infinitesimal analytic transformations represented by finite series* is obtained in this way if, as the subspace \(D\), one takes \(\sum_{i=-2}^{-\infty}\widetilde U_i\). In this case the subspaces \(U_i\) \((i\ge0)\) consist of all \(i+1\)-linear symmetric operators.

The finite-dimensional simple Lie algebra of the series \(B_n\) arises if, as the subspace \(D\), one takes \(\sum_{i=-3}^{-\infty}\widetilde U_i\). In this case \(U_i=0\) for \(i\ge3\), \(U_0\) consists of all linear operators, \(U_1\) of all bilinear operators of the form \(\varphi(y)x\), where \(\varphi(y)\) is an arbitrary linear function, and, finally, \(U_2\) of all trilinear operators of the form \(f(x,z)y\), where \(f(x,z)\) is an arbitrary bilinear skew-symmetric function.

In order to obtain a Lie algebra isomorphic to the infinite-dimensional simple Lie algebra of all infinitesimal transformations of a \((2n+1)\)-dimensional space preserving a linear differential form

* The latter restriction arises because the elements of a graded algebra, by definition, are finite sums of vectors from the subspaces \(U_i\).

\[ dy+\sum_{i=1}^n (x_i dx_i' - x_i' dx_i) \]
and represented by finite series (see the footnote on p. 37), one must take as \(\widehat U_{-1}\) the \(2n\)-dimensional space with the given nondegenerate skew-symmetric form \((x,y)\), and as the subspace \(D\) the subspace
\[ P+\sum_{i=-3}^{-\infty}\widehat U_i, \]
where \(P\subset \widehat U_{-2}\) consists of the elements \(a\circ b\) for which \((a,b)=0\). In this case the subspaces \(U_i\) \((i\geq 0)\) consist of all symmetric \((i+1)\)-linear operators satisfying the condition
\[ (A(x_1,\ldots,x_i,z),y)+(z,A(x_1,x_2,\ldots,x_i,y))=0, \]
and of all operators of the form
\[ \sum_{l>k}^{i+1}(x_k,x_l)B_{i-1}(x_1,\ldots,\widehat{x}_k,\ldots,\widehat{x}_l,\ldots,x_{i+1}) +\sum_{k-1}^{i+1}F(x_1,\ldots,\widehat{x}_k,\ldots,x_{i+1})x_k, \]
where \(B_{i-1}\) \((i\geq 2)\) is an operator of the same type as \(A\), and
\[ F(x_1,\ldots,x_i)=(B_{i-1}(x_1,\ldots,x_{i-1}),x_i); \]
moreover, for \(i=0,1,2\) there are added respectively the following operators: the identity operator \(E\), all bilinear operators of the form
\[ (x,y)a-(x,a)y-(y,a)x \]
and the trilinear operator
\[ (y,z)x-(y,x)-(z,x)y. \]

We shall henceforth restrict ourselves to graded Lie algebras for which, along with (1), the analogous condition
\[ [U_1,U_k]=U_{1+k}\quad (k>0). \tag{6} \]
is satisfied.

With every linear Lie algebra \(G_0\) one can associate a certain special graded Lie algebra \(G\) satisfying conditions (1) and (6). To construct the algebra \(G\), take for \(G_1\) the space generated by linear combinations of bilinear operators of the form \(f(x)A(y)\), where \(f(x)\) is a linear function on \(U_{-1}\) and \(A\in G_0\). Next put \(G_{k+1}=[G_1,G_k]\), where the commutator is defined by formula (3); then
\[ G^* = U_- + G_0 + G_1+\cdots \]
is a graded subalgebra of the algebra \(\widehat U\). Consider the ideal \(D\) of the subalgebra \(\widehat U_-\), generated by the elements \(a\) for which \([a,G_1]=0\). This ideal \(D\) is an ideal of the algebra \(G^*\), and
\[ G=G^*/D. \]

Theorem 3. The algebra \(G\) is an algebra without graded ideals. All other graded algebras with the given linear subalgebra \(G_0\), satisfying conditions (1) and (6) and containing no graded ideals, are obtained from the algebra \(G\) in the following way. To each subspace \(G_1'\subset G_1\) invariant with respect to \(G_0\), for which \([U_{-1},G_1']=G_0\), assign the algebra
\[ G=G_-'+G_+', \]
where \(G_+'\) is the subalgebra of the algebra \(G_+\) generated by \(G_1'\), and
\[ G_-'=\widehat U_-/D'; \]
here \(D'\) denotes the ideal generated by the elements
\[ \{a\in\widehat U_-:[a,G_1']=0\}. \]

Let us note that if condition (6) is discarded, then there are considerably more graded algebras with the given \(G_0\) which contain no graded ideals. In this case, instead of the algebra \(G\), one should consider the graded algebra
\[ \widehat G=\widehat G_-+\widehat G_+, \]
where
\[ \widehat G_1=G_1,\qquad \widehat G_k=\operatorname{Hom}(U_{-1},\widehat G_{k-1}) \]
and
\[ \widehat G_-=\widehat U_-/D, \]
where \(D\) is the ideal of \(\widehat U_-\) generated by the set
\[ \{a\in\widehat U_-:[a,\widehat G_k]=0,\ k=1,2,\ldots\}. \]

Theorem 4. If among the elements of \(U_0\) of the graded algebra \(U\) there is an element \(a\) such that \([a,x]=x\) for all \(x\in U_{-1}\), then every ideal of the algebra \(U\) is graded.

Corollary. For every linear Lie algebra \(G_0\) containing the identity operator, there exists at most one graded simple algebra for which \(U_0=G_0\).

We now note one special case of Theorem 3. Let \(U_{-1}\) be an \(n\)-dimensional space with basis \(e_1, e_2, \ldots, e_n\). As \(G_0\) take the commutative linear algebra consisting of all transformations of the form

\[ \sum_{i=1}^{n} x_i e_i \to \sum_{i=1}^{n} a_i x_i e_i, \]

and as \(G_1'\) take the subspace of bilinear operators

\[ x^i\left(\sum_{j=1}^{n} a_{ij} y_j e_j\right), \]

where \(a_{ij}\) are prescribed numbers \((i=1,2,\ldots,n)\). The matrix \(a_{ij}\), in which the elements of each row are given up to a common factor, is called in the theory of finite-dimensional simple Lie algebras the Cartan matrix. From Theorems 3 and 4 it follows that

Theorem 5. The graded Lie algebra determined by the Cartan matrix \((a_{ij})\) is simple if and only if the matrix \((a_{ij})\) is nonsingular and has no invariant subspaces with a basis consisting of coordinate vectors.

In conclusion we make two remarks. Every simple graded Lie algebra satisfying conditions (1) and (6) is uniquely determined by the triple of subspaces \(U_{-1}, U_0, U_1\) with the prescribed commutation operations \([U_{-1}, U_1]\), \([U_0, U_{-1}]\), \([U_0, U_1]\). If, however, one restricts oneself only to condition (1), then there may exist infinitely many simple algebras with one and the same triple \(U_{-1}, U_0, U_1\). An example is the series of simple graded algebras for which (see Theorem 2) the subalgebra \(U_{-}\) is a free Lie algebra, the subspaces \(U_i\) \((i=0,1,2,\ldots,k-1)\) consist of all \((i+1)\)-linear symmetric operators, and the subspaces \(U_i\) \((i \ge k)\) consist of all \((i+1)\)-linear operators symmetric in the last \(k\) arguments.

The exceptional simple Lie algebras may be represented as elements of infinite series of simple graded algebras. For example, if by \(\hat C_n\) one denotes the graded algebra with \(n\)-dimensional subspace \(U_{-1}\) which arises (see Theorem 2) if, as the subspace \(D\), one takes the ideal generated by elements of the form \(b \circ a \circ d \circ a\), and as \(U_+\) the subalgebra of the algebra \(U_+\) consisting of elements carrying the subspace \(D\) into itself, then for \(n=2\) one obtains the exceptional Lie algebra \(G_2\), and for \(n>3\) infinite-dimensional simple graded Lie algebras.

The author expresses his gratitude to B. E. Weisfeiler, A. I. Sirota, and A. S. Solodovnikov for useful discussions.

All-Union Correspondence
Financial-Economic Institute

Received
15 V 1967